linking l -chu correspondences and completely lattice l...

Linking L-Chu correspondences andcompletely lattice L-ordered sets

Ondrej Krıdlo1 and Manuel Ojeda-Aciego2

1 University of Pavol Jozef Safarik, Kosice, Slovakia?

2 Dept. Matematica Aplicada, Univ. Malaga, Spain??

Abstract. Continuing our categorical study of L-fuzzy extensions offormal concept analysis, we provide a representation theorem for thecategory of L-Chu correspondences between L-formal contexts and provethat it is equivalent to the category of completely lattice L-ordered sets.

1 Introduction

This paper deals with an extremely general form of Formal Concept Analysis(FCA) based on categorical constructs and L-fuzzy sets. FCA has become anextremely useful theoretical and practical tool for formally describing structuraland hierarchical properties of data with “object-attribute” character, and thisapplicability justifies the need of a deeper knowledge of its underlying mecha-nisms: and one important way to obtain this extra knowledge turns out to bevia generalization and abstraction.

Several approaches have been presented for generalizing the framework andthe scope of formal concept analysis and, nowadays, one can see works whichextend the theory by using ideas from fuzzy set theory, rough set theory, orpossibility theory [1, 10,18,20–22,24].

Concerning applications of fuzzy formal concept analysis, one can see papersranging from ontology merging [9], to applications to the Semantic Web by usingthe notion of concept similarity or rough sets [11,12], and from noise control indocument classification [19] to the development of recommender systems [7].

We are concerned in this work with the category L-ChuCors, built on topof several fuzzy extensions of the classical concept lattice, mainly introduced byBelohlavek [3,5,6], who extended the underlying interpretation on classical logicto the more general framework of L-fuzzy logic [13].

The categorical treatment of morphisms as fundamental structural proper-ties has been advocated by [17] as a means for the modelling of data translation,communication, and distributed computing, among other applications. Our ap-proach broadly continues the research line which links the theory of Chu spaceswith concept lattices [25] but, particularly, is based on the notion of Chu corre-spondences between formal contexts developed by Mori in [23]. Previous work

? Partially supported by grant VEGA 1/0832/12 and APVV-0035-10.?? Partially supported by Spanish Ministry of Science and FEDER funds through

project TIN09-14562-C05-01 and Junta de AndalucAa project P09-FQM-5233.

c© Laszlo Szathmary, Uta Priss (Eds.): CLA 2012, pp. 233–244, 2012.ISBN 978–84–695–5252–0, Universidad de Malaga (Dept. Matematica Aplicada)

in this categorical approach has been developed by the authors in [14, 16]. Thecategory L-ChuCors is formed by considering the class of L-contexts as objectsand the L-fuzzy Chu correspondences as arrows between objects. Recently, theauthors developed a further abstraction [15] aiming at formally describing struc-tural properties of intercontextual relationships of L-contexts.

The main result in this work is a constructive proof of the equivalence be-tween the categories of L-formal contexts and L-Chu correspondences and thatof completely lattice L-ordered sets and their corresponding morphisms. In orderto obtain a reasonably self-contained document, Section 2 introduces the basicdefinitions concerning the L-fuzzy extension of formal concept analysis, as wellas those concerning L-Chu correspondences; then, the categories associated toL-formal contexts and L-CLLOS are defined in Section 3 and, finally, the proofof equivalence is in Section 4.

2 Preliminaries

2.1 Basics of L-fuzzy FCA

Definition 1. An algebra 〈L,∧,∨,⊗,→, 0, 1〉 is said to be a complete resid-uated lattice if

– 〈L,∧,∨, 0, 1〉 is a complete lattice with the least element 0 and the greatestelement 1,

– 〈L,⊗, 1〉 is a commutative monoid,– ⊗ and→ are adjoint, i.e. a⊗b ≤ c if and only if a ≤ b→ c, for all a, b, c ∈ L,

where ≤ is the ordering in the lattice generated from ∧ and ∨.

Definition 2. Let L be a complete residuated lattice, an L-fuzzy context is atriple 〈B,A, r〉 consisting of a set of objects B, a set of attributes A and an L-fuzzy binary relation r, i.e. a mapping r : B×A→ L, which can be alternativelyunderstood as an L-fuzzy subset of B ×A.

Definition 3. Consider an L-fuzzy context 〈B,A, r〉. Mappings ↑ : LB → LA

and ↓ : LA → LB can be defined for every f ∈ LB and g ∈ LA as follows:

↑ (f)(a) =∧

o∈B

(f(o)→ r(o, a)

)↓ (g)(o) =

∧

a∈A

(g(a)→ r(o, a)

)

Definition 4. An L-fuzzy concept is a pair 〈f, g〉 such that ↑ (f) = g and↓ (g) = f . The first component f is said to be the extent of the concept, whereasthe second component g is the intent of the concept.

The set of all L-fuzzy concepts associated to a fuzzy context 〈B,A, r〉 will bedenoted as L-FCL(B,A, r).

An ordering between L-fuzzy concepts is defined as follows: 〈f1, g1〉 ≤ 〈f2, g2〉if and only if f1 ⊆ f2 (f1(o) ≤ f2(o) for all o ∈ B) if and only if g1 ⊇ g2(g1(o) ≥ g2(o) for all a ∈ A).

234 Ondrej Krıdlo and Manuel Ojeda-Aciego

Theorem 1 (See [5]). The poset (L-FCL(B,A, r),≤) is a complete latticewhere

∧

j∈J〈fj , gj〉 =

⟨ ∧

j∈Jfj , ↑

( ∧

j∈Jfj)⟩

∨

j∈J〈fj , gj〉 =

⟨↓( ∧

j∈Jgj),

∧

j∈Jgj

⟩

Moreover a complete lattice V = 〈V,≤〉 is isomorphic to L-FCL(B,A, r) iffthere are mappings γ : B × L → V and µ : A × L → V , such that γ(B × L)is∨

-dense and µA× L is∧

-dense in V, and (k ⊗ l) ≤ r(o, a) is equivalent toγ(o, k) ≤ µ(a, l) for all o ∈ B, a ∈ A and k, l ∈ L.

Belohlavek has extended the fundamental theorem of concept lattices byDedekind-MacNeille completion in fuzzy settings by using the notions of L-equality and L-ordering. All the definitions and related constructions given untilthe end of the section are from [6].

Definition 5. A binary L-relation ≈ on X is called an L-equality if it satisfies

1. (x ≈ x) = 1, (reflexivity),2. (x ≈ y) = (y ≈ x), (symmetry),3. (x ≈ y)⊗ (y ≈ z) ≤ (x ≈ z), (transitivity),4. (x ≈ y) = 1 implies x = y

L-equality is a natural generalization of the classical (bivalent) notion.

Definition 6. An L-ordering (or fuzzy ordering) on a set X endowed with anL-equality relation ≈ is a binary L-relation � which is compatible w.r.t. ≈ (i.e.f(x)⊗ (x ≈ y) ≤ f(y), for all x, y ∈ X) and satisfies

1. x � x = 1, (reflexivity),2. (x � y) ∧ (y � x) ≤ (x ≈ y), (antisymmetry),3. (x � y)⊗ (y � z) ≤ (x � z), (transitivity).

If � is an L-order on a set X with an L-equality ≈, we call the pair 〈〈X,≈〉 �〉an L-ordered set.

Clearly, if L = 2, the notion of L-order coincides with the usual notion of(partial) order.

Definition 7. An L-set f ∈ LX is said to be an L-singleton in 〈X,≈〉 if it iscompatible w.r.t. ≈ and the following holds:

1. there exists x0 ∈ X with f(x0) = 12. f(x)⊗ f(y) ≤ (x ≈ y), for all x, y ∈ X.

Definition 8. For an L-ordered set 〈〈X,≈〉 �〉 and f ∈ LX we define the L-setsinf(f) and sup(f) in X by

Linking L-Chu correspondences and completely lattice L-ordered sets 235

1. inf(f)(x) = (L(f))(x) ∧ (UL(f))(x)2. sup(f)(x) = (U(f))(x) ∧ (LU(f))(x)

where

– L(f)(x) =∧

y∈X(f(y)→ (x � y)

)

– U(f)(x) =∧

y∈X(f(y)→ (y � x)

)

inf(f) and sup(f) are called infimum or supremum, respectively.

Definition 9. An L-ordered set 〈〈X,≈〉 �〉 is said to be completely latticeL-ordered set if for any f ∈ LX both sup(f) and inf(f) are ≈-singletons.

By proving of all the following lemmas some of the properties of residuatedlattices are used. All details could be found in [4]. Some of needed properties arelisted below.

1. (k → (l→ m)) = ((k ⊗ l)→ m) = ((l ⊗ k)→ m) = (l→ (k → m))2. k → ∧

i∈I mi =∧

i∈I(k → mi)3. (

∨i∈I mi)→ k =

∧i∈I(mi → k)

Lemma 1. For any pair of L-concepts 〈fi, gi〉 ∈ L-FCL(B,A, r) (i ∈ {1, 2}) ofany L-context 〈B,A, r〉 the following equality holds.

∧

o∈B

(f1(o)→ f2(o)

)=∧

a∈A

(g2(a)→ g1(a)

)

Proof.

∧

o∈B

(f1(o)→ f2(o)

)=∧

o∈B

(f1(o)→ ↓ (g2)(o)

)

=∧

o∈B

(f1(o)→

∧

a∈A

(g2(a)→ r(o, a)

))

=∧

a∈A

(g2(a)→

∧

o∈B

(f1(o)→ r(o, a)

))

=∧

a∈A

(g2(a)→ ↑ (f1)(a)

)

=∧

a∈A

(g2(a)→ g1(a)

)ut

Definition 10. We define an L-equality ≈ and L-ordering � on the set of for-mal concepts L-FCL(C) of L-context C as follows:

1. 〈f1, g1〉 � 〈f2, g2〉 =∧

o∈B(f1(o)→ f2(o)

)=∧

a∈A(g2(a)→ g1(a)

)

2. 〈f1, g1〉 ≈ 〈f2, g2〉 =∧

o∈B(f1(o)↔ f2(o)

)=∧

a∈A(g2(a)↔ g1(a)

)

where k ↔ m is defined as (k → m) ∧ (m→ k) for any k,m ∈ L.


Definition 11. Let C = 〈B,A, r〉 be an L-fuzzy formal context and γ be an L-set from LL-FCL(C). We define L-sets of objects and attributes

⋃B γ and

⋃A γ,

respectively, as follows:

1. (⋃

B γ)(o) =∨

〈f,g〉∈L-FCL(C)

(γ(〈f, g〉)⊗ f(o)), for o ∈ B

2. (⋃

A γ)(a) =∨

〈f,g〉∈L-FCL(C)

(γ(〈f, g〉)⊗ g(a)), for a ∈ A

Theorem 2. Let C = 〈B,A, r〉 be an L-context. 〈〈L-FCL(C),≈〉,�〉 is a com-pletely lattice L-ordered set in which infima and suprema can be described asfollows: for an L-set γ ∈ LL−FCL(C) we have:

1 inf(γ) ={⟨↓(⋃

A

γ), ↑↓

(⋃

A

γ)⟩}

1 sup(γ) ={⟨↓↑(⋃

B

γ), ↑(⋃

B

γ)⟩}

Moreover a completely lattice L-ordered set V = 〈〈V,≈〉,�〉 is isomorphicto 〈〈L-FCL(〈B,A, r〉),≈1〉,�1〉 iff there are mappings γ : B × L → V and µ :A×L→ V , such that γ(B×L) is {0, 1}-supremum dense and µ(A×L) is {0, 1}-infimum dense in V, and ((k ⊗ l) → r(o, a)) = (γ(o, k) � µ(a, l)) for all o ∈ B,a ∈ A and k, l ∈ L. In particular, V is isomorphic to 〈〈L-FCL(V, V,�),≈1〉,�1〉.

2.2 L-Chu correspondences

Definition 12. Consider two L-fuzzy contexts Ci = 〈Bi, Ai, ri〉, (i = 1, 2), thenthe pair ϕ = (ϕL, ϕR) is called a correspondence from C1 to C2 if ϕL and ϕR

are L-multifunctions, respectively, from B1 to B2 and from A2 to A1 (that is,ϕL : B1 → LB2 and ϕR : A2 → LA1).

The L-correspondence ϕ is said to be a weak L-Chu correspondence ifthe equality

∧

a1∈A1

(ϕR(a2)(a1)→ r1(o1, a1)) =∧

o2∈B2

(ϕL(o1)(o2)→ r2(o2, a2)) (1)

holds for all o1 ∈ B1 and a2 ∈ A2.A weak Chu correspondence ϕ is an L-Chu correspondence if ϕL(o1) is an

L-set of objects closed in C2 and ϕR(a2) is an L-set of attributes closed in C1 forall o1 ∈ B1 and a2 ∈ A2. We will denote the set of all L-Chu correspondencesfrom C1 to C2 by L-ChuCors(C1, C2).

Definition 13. Given a mapping $ : X → LY , we define $+ : LX → LY forall f ∈ LX by $+(f)(y) =

∨x∈X

(f(x)⊗$(x)(y)

).

3 Introducing the relevant categories

3.1 The category L-ChuCors

– objects L-fuzzy formal contexts


– arrows L-Chu correspondences– identity arrow ι : C → C of L-context C = 〈B,A, r〉• ιL(o) =↓↑ (χo), for all o ∈ B• ιR(a) =↑↓ (χa), for all a ∈ A

– composition ϕ2 ◦ ϕ1 : C1 → C3 of arrows ϕ1 : C1 → C2, ϕ2 : C2 → C3

(Ci = 〈Bi, Ai, ri〉, i ∈ {1, 2})• (ϕ2 ◦ϕ1)L : B1 → LB3 defined as (ϕ2 ◦ϕ1)L(o1) = ↓3↑3

(ϕ2L+(ϕ1L(o1))

)

where

ϕ2L+(ϕ1L(o1))(o3) =∨

o2∈B2

ϕ1L(o1)(o2)⊗ ϕ2L(o2)(o3)

• and (ϕ2◦ϕ1)R : A3 → LA1 defined as (ϕ2◦ϕ1)R(a3) = ↑1↓1(ϕ1R+(ϕ2R(a3))

)

where

ϕ1R+(ϕ2R(a3))(a1) =∨

a2∈A2

ϕ2R(a3)(a2)⊗ ϕ1R(a2)(a1)

All details about definition of the category L-ChuCors could be found in [15].

3.2 Category L-CLLOS

Here we define another category

Objects are completely lattice L-ordered sets (in short, L-CLLOS) i.e. our ob-jects will be represented as V = 〈〈V,≈〉,�〉

Arrows are pairs of mappings between two L- CLLOSs i.e. 〈s, z〉 between V1and V2, such that:1. s : V1 → V2,2. z : V2 → V1,3. (s(v1) �2 v2) = (v1 �1 z(v2)), for all (v1, v2) ∈ V1 × V2.

Identity arrow of 〈〈V,≈〉,�〉 is a pair of identity morphisms 〈idV , idV 〉Composition of arrows is based on composition of mappings: consider

two arrows 〈si, zi〉 : Vi → Vi+1, where i ∈ {1, 2}. Composition is definedas follows:

〈s2, z2〉 ◦ 〈s1, z1〉 = 〈s2 ◦ s1, z1 ◦ z2〉.Thus, given a pair of two arbitrary elements (v1, v3) ∈ V1 × V3 then:

((s2 ◦ s1)(v1) �3 v3

)=(s2(s1(v1)) �3 v3

)

=(s1(v1) �2 z2(v3)

)

=(v1 �1 z1(z2(v3))

)

=(v1 �1 (z1 ◦ z2)(v3)

)

Associativity of composition follows trivially because of the associativ-ity of composition of mappings between sets.


4 The categories L-ChuCors and L-CLLOS are equivalent

In this section we start to build the equivalence by introducing a functor Γ fromL-ChuCors to L-CLLOS in the following way:

1. Γ (C) = 〈〈L-FCL(C),≈〉,�〉 for any L-context C will be its L-concept L-CLLOS.

2. Γ (ϕ) = 〈ϕ∨, ϕ∧〉. To any L-Chu correspondence ϕ ∈ L-ChuCors(C1, C2),Γ (ϕ) will be a pair of mappings 〈ϕ∨, ϕ∧〉 defined as follows:– ϕ∨

(〈f1, g1〉

)=⟨↓2↑2

(ϕL+(f1)

), ↑2

(ϕL+(f1)

)⟩

– ϕ∧(〈f2, g2〉

)=⟨↓1(ϕR+(g2)

), ↑1↓1

(ϕR+(g2)

)⟩where 〈fi, gi〉 ∈ L-FCL(Ci) for i ∈ {1, 2}.

Lemma 2. Γ (ϕ) ∈ L-CLLOS(Γ (C1), Γ (C2)) for any ϕ ∈ L-ChuCors(C1, C2).

Proof. Consider two arbitrary L-concepts 〈fi, gi〉 of 〈〈L-FCL(Ci),≈i〉,�i〉 fori ∈ {1, 2}, such that Ci = 〈Bi, Ai, ri〉.ϕ∨(〈f1, g1〉

)�2 〈f2, g2〉=⟨↓2↑2

(ϕL+(f1)

), ↑2

(ϕL+(f1)

)⟩�2 〈f2, g2〉

=∧

a2∈A2

(g2(a2)→ ↑2

(ϕL+(f1)

)(a2)

)

=∧

a2∈A2

(g2(a2)→

∧

o2∈B2

( ∨

o1∈B1

(ϕL(o1)(o2)⊗ f1(o1)

)→ r2(o2, a2)

))

=∧

a2∈A2

∧

o1∈B1

(g2(a2)→

(f1(o1)→

∧

o2∈B2

(ϕL(o1)(o2)→ r2(o2, a2)

)))

=∧

a2∈A2

∧

o1∈B1

(f1(o1)→

(g2(a2)→

∧

a1∈A1

(ϕR(a2)(a1)→ r1(o1, a1)

)))

=∧

o1∈B1

(f1(o1)→

∧

a1∈A1

( ∨

a2∈A2

(ϕR(a2)(a1)⊗ g2(a2)

)→ r1(o1, a1)

))

=∧

o1∈B1

(f1(o1)→ ↓1

(ϕR+(g2)

)(o1)

)

= 〈f1, g1〉 �1

⟨↓1(ϕR+(g2)

), ↑1↓1

(ϕR+(g2)

)⟩

= 〈f1, g1〉 �1 ϕ∧(〈f2, g2〉

)

utLemma 3. For the identity arrow ι ∈ L-ChuCors(C,C) of any L-context C =〈B,A, r〉, Γ (ι) is the identity arrow from L-CLLOS(Γ (C), Γ (C)).

Proof. Consider any L-concept 〈f, g〉 from L-FCL(C).

↑(ιL+(f)

)(a) =

∧

o∈B

(ιL+(f)(o)→ r(o, a)

)

=∧

o∈B

( ∨

b∈B

(ιL(b)(o)⊗ f(b)

)→ r(o, a)

)


=∧

o∈B

∧

b∈B

((ιL(b)(o)⊗ f(b)

)→ r(o, a)

)

=∧

o∈B

(f(b)→

∧

b∈B

(ιL(b)(o)→ r(o, a)

)

=∧

b∈B

(f(b)→ ↑↓↑

(χb

)(a))

=∧

b∈B

(f(b)→ r(b, a)

)= ↑ (f)(a)

Therefore, we have ι∨(〈f, g〉) = 〈f, g〉. The proof for ι∧ is similar. ut

Lemma 4. Consider arbitrary ϕi ∈ L-ChuCors(Ci, Ci+1) for i ∈ {1, 2} and anyelement o1 ∈ B1 and g3 ∈ LA3 . Then

↓1(ϕ1R+

(ϕ2R+(g3)

))(o1) = ↓1

(ϕ1R+

(↑2↓2

(ϕ2R+(g3)

)))(o1).

Proof.

↓1(ϕ1R+

(ϕ2R+(g3)

))(o1)

=∧

a1∈A1

(ϕ1R+

(ϕ2R+(g3)

)(a1)→ r1(o1, a1)

)

=∧

a1∈A1

( ∨

a2∈A2

(ϕ1R(a2)(a1)⊗ ϕ2R+(g3)(a2)

)→ r1(o1, a1)

)

=∧

a2∈A2

(ϕ2R+(g3)(a2)→

∧

a1∈A1

(ϕ1R(a2)(a1)→ r1(o1, a1)

))

=∧

a2∈A2

(ϕ2R+(g3)(a2)→

∧

o2∈B2

(ϕ1L(o1)(o2)→ r2(o2, a2)

))

=∧

o2∈B2

(ϕ1L(o1)(o2)→

∧

a2∈A2

(ϕ2R+(g3)(a2)→ r2(o2, a2)

))

=∧

o2∈B2

(ϕ1L(o1)(o2)→ ↓2↑2↓2

(ϕ2R+(g3)

)(o2)

)

by applying the same chain of modifications in opposite way we will obtain

= ↓1(ϕ1R+

(↑2↓2

(ϕ2R+(g3)

)))(o1)

ut

Lemma 5. Mapping Γ is closed under arrow composition.

Proof. Consider ϕi ∈ L-ChuCors(Ci, Ci+1) for i ∈ {1, 2}. Let 〈fi, gi〉 ∈ L-FCL(Ci)be an arbitrary L-context for all i ∈ {1, 3}. Recall that

1. Γ (ϕ2 ◦ ϕ1) =⟨(ϕ2 ◦ ϕ1)∨, (ϕ2 ◦ ϕ1)∧

⟩

2. Γ (ϕ2) ◦ Γ (ϕ1) =⟨ϕ2∨ ◦ ϕ1∨, ϕ1∧ ◦ ϕ2∧

⟩


The proof will be based on equality of corresponding elements of the previouspairs: only one part will be proved, the other one is similar.

(ϕ1∧ ◦ ϕ2∧)(〈f3, g3〉

)= ϕ1∧

(ϕ2∧

(〈f3, g3〉

))

=⟨↓1 (ϕ1R+(↑2↓2 (ϕ2R+(g3)))), ↑1↓1 (ϕ1R+(↑2↓2 (ϕ2R+(g3))))

⟩

by lemma 4 we have

=⟨↓1 (ϕ1R+(ϕ2R+(g3))), ↑1↓1 (ϕ1R+(ϕ2R+(g3)))

⟩= ?

↓1 (ϕ1R+(ϕ2R+(g3)))(o1) =

=∧

a1∈A1

( ∨

a2∈A2

(ϕ1R(a2)(a1)⊗ ϕ2R+(g3)(a2))→ r1(o1, a1))

=∧

a1∈A1

( ∨

a2∈A2

(ϕ1R(a2)(a1)⊗∨

a3∈A3

(ϕ2R(a3)(a2)⊗ g3(a3)))→ r1(o1, a1))

=∧

a1∈A1

( ∨

a3∈A3

(ϕ1R+(ϕ2R(a3))(a1)⊗ g3(a3)

)→ r1(o1, a1)

)

=∧

a3∈A3

(g3(a3)→

∧

a1∈A1

(ϕ1R+(ϕ2R(a3))(a1)→ r1(o1, a1)

))

=∧

a3∈A3

(g3(a3)→↓1↑1↓1

(ϕ1R+(ϕ2R(a3))

)(o1)

)

=∧

a3∈A3

(g3(a3)→↓1

((ϕ2 ◦ ϕ1)R(a3)

)(o1)

)

=∧

a3∈A3

(g3(a3)→

∧

a1∈A1

((ϕ2 ◦ ϕ1)R(a3)(a1)→ r1(o1, a1)

))

=∧

a1∈A1

∧

a3∈A3

((g3(a3)⊗ (ϕ2 ◦ ϕ1)R(a3)(a1)

)→ r1(o1, a1)

)

=∧

a1∈A1

( ∨

a3∈A3

(g3(a3)⊗ (ϕ2 ◦ ϕ1)R(a3)(a1))→ r1(o1, a1))

=↓1((ϕ2 ◦ ϕ1)R+(g3)

)(o1)

? =⟨↓1 ((ϕ2 ◦ ϕ1)R+(g3)), ↑1↓1 ((ϕ2 ◦ ϕ1)R+(g3))

⟩

=(ϕ2 ◦ ϕ1)∧(〈f3, g3〉)

utProposition 1. Γ is a functor from L-ChuCorrs to L-CLLOS.

Proof. Straightforward from the previous lemmas. utWe continue by showing that the previous functor satisfies the conditions to

define a categorical equivalence, characterized by the following result:


Theorem 3 (See [2]). The following conditions on a functor F : C → D areequivalent:

– F is an equivalence of categories.– F is full and faithful and ”essentially surjective” on objects: for every D ∈ D

there is some C ∈ C such that F (C) ∼= D.

Let us recall the definition of the notions required by the previous theorem:

Definition 14.

1. A functor F : C → D is faithful if for all objects A,B of a category C, themap FA,B : HomC(A,B)→ HomD(F (A), F (B)) is injective.

2. Similarly, F is full if FA,B is always surjective.

In our cases, for proving fullness and faithfulness of the functor Γ we needto prove surjectivity and injectivity of the mapping

ΓC1,C2: L-ChuCors(C1, C2)→ L-CLLOS(Γ (C1), Γ (C2))

for any two L-contexts C1 and C2. This will be done in the forthcoming lemmas.

Lemma 6. Γ is full.

Proof. The point of the proof is to show that given any arrow 〈s, z〉 from the setL-CLLOS(Γ (C1), Γ (C2)) there exists an L-Chu correspondence ϕ〈s,z〉 from theset L-ChuCors(C1, C2), for any two L-contexts Ci = 〈Bi, Ai, ri〉 for i = {1, 2}.Let us define the following mappings:

– ϕ〈s,z〉L (o1) = Ext

(s(〈↓1↑1 (χo1), ↑1 (χo1)〉

))

– ϕ〈s,z〉R (a2) = Int

(z(〈↓2 (χa2

), ↑2↓2 (χa2)〉))

↑2(ϕ〈s,z〉L (o1)

)(a2) =

∧

o2∈B2

(ϕ〈s,z〉L (o1)(o2)→ r2(o2, a2)

)

=∧

o2∈B2

(Ext(s(〈↓1↑1 (χo1), ↑1 (χo1)〉))(o2)→ ↓2 (χa2)(o2)

)

= s(〈↓1↑1 (χo1), ↑1 (χo1)〉

)�2 〈↓2 (χa2

), ↑2↓2 (χa2)〉

= 〈↓1↑1 (χo1), ↑1 (χo1)〉 �1 z(〈↓2 (χa2

), ↑2↓2 (χa2)〉)

=∧

a1∈A1

(Int(z(〈↓2 (χa2

), ↑2↓2 (χa2)〉))(a1)→ ↑1 (χo1)(a1)

)

=∧

a1∈A1

(ϕ〈s,z〉R (a2)(a1)→ r1(o1, a1)

)

= ↓1(ϕ〈s,z〉R (a2)

)(o1)

So ϕ〈s,z〉 ∈ L-ChuCors(C1, C2) and ΓC1,C2 is surjective, hence Γ is full. ut


Lemma 7. Γ is faithfull

Proof. Now the point is to prove the injectivity of ΓC1,C2 .Consider two L-Chu correspondences ϕ1, ϕ2 from L-ChuCors(C1, C2) such

that ϕ1 6= ϕ2, and let us fix the pair (o1, a2) ∈ B1 ×A2, such that

↑2(ϕ1L(o1)

)(a2) = ↓1

(ϕ1R(a2)

)(o1) 6= ↑2

(ϕ2L(o1)

)(a2) = ↓1

(ϕ2R(a2)

)(o1)

Let us assume that either ↓1(ϕ1R(a2)

)(o1) > ↑2

(ϕ2L(o1)

)(a2) or that both

values from L are incomparable, that is equivalent to the following:

↓1(ϕ1R(a2)

)(o1)→ ↑2

(ϕ2L(o1)

)(a2) < 1

Now consider the L-concept 〈↓1 (ϕ1R(a2)), ϕ1R(a2)〉 and let us compare itsimages under the mappings ϕ1∨ and ϕ2∨.

↑2(ϕ2L+

(↓1 (ϕ1R(a2))

))(a2)

=∧

o2∈B2

(ϕ2L+(↓1 (ϕ1R(a2)))(o2)→ r2(o2, a2)

)

=∧

o2∈B2

( ∨

b1∈B1

(ϕ2L(b1)(o2)⊗ ↓1 (ϕ1R(a2))(b1)

)→ r2(o2, a2)

)

=∧

b1∈B1

(↓1 (ϕ1R(a2))(b1)→

∧

o2∈B2

(ϕ2L(b1)(o2)→ r2(o2, a2)

))

=∧

b1∈B1

(↓1 (ϕ1R(a2))(b1)→ ↑2 (ϕ2L(b1))(a2)

)

≤↓1 (ϕ1R(a2))(o1)→ ↑2 (ϕ2L(o1))(a2)

< 1 because of the restriction given above

Similarly, we can obtain:

↑2 (ϕ1L+(↓1 (ϕ1R(a2))))(a2) =

=∧

b1∈B1

(↓1 (ϕ1R(a2))(b1)→ ↑2 (ϕ1L(b1))(a2)

)

=∧

b1∈B1

(↓1 (ϕ1R(a2))(b1)→ ↓1 (ϕ1R(a2))(b1)

)= 1

It means that ϕ1∨(〈↓1 (ϕ1R(a2)), ϕ1R(a2)〉

)6= ϕ2∨

(〈↓1 (ϕ1R(a2)), ϕ1R(a2)〉

)

Hence ϕ1∨(〈↓1 (ϕ1R(a2)), ϕ1R(a2)〉) 6= ϕ2∨(〈↓1 (ϕ1R(a2)), ϕ1R(a2)〉) andϕ1∨ 6= ϕ2∨. So ΓC1,C2

is injective and Γ is faithfull. utProposition 2. The functor Γ is an equivalence functor between L-ChuCorsand L-CLLOS.

Proof. Fullness and faithfulness of Γ is given by previous lemmas. Essential sur-jectivity on objects is ensured by the fact that given any object 〈〈V,≈〉,�〉 ofL-CLLOS there exists an L-context 〈V, V,�〉, such that Γ

(〈V, V,�〉

) ∼= 〈〈V,≈〉,�〉.Hence, we can state that Γ is the functor of equivalence between L-ChuCors andL-CLLOS. ut


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